# Chapter 3 - Section 3.9 - Inverse Trigonometric Functions - Exercises - Page 181: 19

$0$

#### Work Step by Step

$y=\csc^{-1}x$ is the number in $[-\pi/2,0) \cup(0, \pi/2]$ for which $\csc y=x$ $($In terms of sine, $\displaystyle \quad \sin y=\frac{1}{x}$) We want an angle for which sine approaches 0, so the cosecant $\rightarrow\infty)$. The angle (in radians) is $0$, (we observe $[-\pi/2,0) \cup(0, \pi/2]$). Alternatively (if in doubt), we can reach the same conclusion by observing the graph of $y=\csc x$ (also written as $\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{s}\mathrm{c} x$) when $x\rightarrow\infty$. See below.

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